Extracting Embedded Orthogonal Array

Given a (digital) (m−k, m, s)-netN in base b with k ≤ s, a (linear) orthogonal arrayA with parameters OA(bm, s, Sb, k) can be constructed. If N is digital over Fb, A is linear and its dual is a linear [s, s−m, k + 1]-code over Fb.

For the digital/linear case the result is given explicitly for the first time in [1, Section 1] and [2, Section 3], even though the connection between digital nets and linear codes was already pointed out in [3, Remark 7.13], where the special case for digital (0, m, s)-nets and MDS-codes is considered.

The result for arbitrary nets and OAs is first given in [4] (see also [5, Corollary 9]). The special case with k = 2 can already be found in [2, Theorem 1].

Construction

A is formed based on the leading digit in the b-adic expansion of the coordinates of the points of N. More formally, A is obtained from N as

A = {(ηi−1(⌊bxi⌋))i=1,…, s : x ∈ N}

with ηi : Sb↔{0,…, b – 1} denoting arbitrary bijections.

If N is digital over Fb, any k of the first row vectors of the s generator matrices of N are linearly independent and form the generator matrix of a linear OA(bm, s,Fb, k).